Anomalous states of positronium

The energies and wave functions of the two-body Dirac equation for positronium are compared with those of the Pauli approximation and the Bethe-Salpeter equation. The unusual behavior of the ground-state wave function of the Dirac equation is explained in terms of anomalous bound-state solutions.

Figure 1

The Dirac and Pauli radial wave function components (squared) for the ground-state are compared. Large dots correspond to the 399 grid points used for the finite element calculation. Plots are on a log-log scale.

Figure 2

The energies $E_i$ for the anomalous states are found by solving the same equations as in Fig. 1. The finite element grid points ${\rho }_i$ shown here are the same as in Region 1 of Fig. 1. Energies are given analytically by $-1/{\rho }_i$ hartree shown by the solid line. Vertical lines are at the element boundaries.

Figure 3

The anomalous bound-state wave functions in the coordinate representation for the energies in Fig. 2 at the element boundaries. The wave functions are approximately evenly spaced delta functions. The component $(y_{12}^{+}+y_{21}^{+})$ is too small to be seen.

Figure 4

Anomalous bound-state wave functions as in Fig. 3 but calculated in the momentum representation. The component $(y_{12}^{\alpha }+y_{21}^{\alpha })$ is too small to be seen. Analytic results are from (38).

Figure 5

The Dirac radial wave functions in Fig. 1 are compared with analytical results in which the Dirac zero-order ground-state $|{\mathrm{\Psi }}_D{\rangle }^{\left(0\right)}$ is coupled to the anomalous states $|{\mathrm{\Psi }}_S^0,i\rangle$ by the Coulomb potential. Plots are on a log-log scale.